The Axioms of Adaptivity and Least-Squares Finite Element Methods
A talk in the Keine Reihe series by
Carsten Carstensen from HU Berlin
| Abstract: | The least-squares functional is a reliable and efficient error estimator with
global upper and lower bounds and can be a very accurate error estimator for some related
norm as it is often asymptotically exact from Carstensen and Storn (SIAM J. Numer. Anal.
2018).
The piecewise contributions to the global L2 norms serve well as refinement indicators in
adaptive mesh-refining algorithms but the convergence analysis is less well understood.
Those local contributions do not involve an explicit mesh-size factor and hence their reduc-
tion is unclear.
The paper Bringmann, Carstensen, and Park (Numer. Math. 2017) guarantees the plain
convergence for a bulk parameter close to one and that is far away from the arguments
for rate optimality. The axioms of adaptivity in Carstensen, Feischl, Page, and Praetorius
(Comp. Math. Appl. 2014) are not available and an alternative error estimator is derived and
exploited in Carstensen and Park (SIAM J. Numer. Anal. 2015) and enforces a separate
marking strategy with an overall abstract theory by Carstensen and Rabus (SIAM J. Numer.
Anal. 2017).
The presentation discusses on all those aspects for the Laplace, the Stokes and the Lame-
Navier equations as in Bringmann and Carstensen (Numer. Math. 2017) and Bringmann,
Carstensen, and Starke (SIAM J. Numer. Anal. 2018). The norm equivalence gives rise
to separate marking and the question arises whether collective marking is excluded from
adaptive least-squares FEMs. A first negative answer from Carstensen (Math. Comp. 2019
electr.) to this question concludes the presentation. |